Averages and spread

• Median: order the values; with $n$ values it is the $\frac{n+1}{2}$th. With an even number, it is halfway between the middle two.
• Mode: the most frequent value. There can be more than one, or none.
• Range: largest minus smallest — a single number, not “3 to 17”.

Reverse questions come up often: “the mean of 5 numbers is 12; a sixth number is added and the mean becomes 13”. Work with totals: $5\times 12=60$ and $6\times 13=78$, so the new number is 18.

Multiply each value by its frequency, add those up, and divide by the total frequency — not by the number of rows. The median is the value at position $\frac{n+1}{2}$ counting through the cumulative frequencies; the mode is the value with the highest frequency.

When the data is in classes such as $10<t\le 20$, the exact values are unknown, so each class is represented by its midpoint:

It is an estimate because it assumes every value sits at the middle of its class. The modal class is the class with the highest frequency; for unequal class widths that means the highest frequency density, which is where this meets histograms.

The lower quartile is a quarter of the way through the ordered data and the upper quartile three-quarters of the way. The interquartile range is upper quartile minus lower quartile — the spread of the middle half, which, unlike the range, ignores extreme values. For grouped data they are read from a cumulative frequency curve at $\frac{1}{4}$ and $\frac{3}{4}$ of the total frequency.

“Compare the distributions” wants two sentences: one about an average and one about spread, each naming the numbers and saying what they mean in context. For example: “Class A’s median score was higher (34 against 29), so on average Class A did better. Class B’s interquartile range was smaller (6 against 11), so Class B’s scores were more consistent.”

Use the median and interquartile range when there are extreme values; the mean and range are pulled about by them.

1. 1.Dividing by the number of classes instead of the total frequency.

   In the example above that gives $1110÷4$. The divisor is $\sum f=50$.

2. 2.Using the class boundary instead of the midpoint.

   The class $10<t\le 20$ is represented by 15, not 10 or 20.

3. 3.Giving the frequency as the mode.

   The mode is the value (or class) that occurs most, not how many times it occurs.

4. 4.Finding the median without ordering the data.

   Order first. In a frequency table the values are already ordered — count through the cumulative frequency.

5. 5.Comparing without context.

   “The median of A is higher” earns less than “A’s median time is higher, so A’s students generally took longer”.